form „¦ = dx§dy, this action is Hamiltonian but not Poisson.

7. Verify the claim made in the proof of Theorem 1 that the following identity holds for

all a ∈ G, all y ∈ g, and all m ∈ M:

»a »— (y)(m) = »— Ad(a)y (a · m).

8. Here are a few mechanical exercises that turn out to be useful in calculations:

(i) Show that if »i : G — Mi ’ Mi for i = 1, 2 are Poisson actions on symplectic

manifolds (Mi , „¦i ) with corresponding momentum mappings µi : Mi ’ g— , then

the induced product action of G on M = M1 — M2 (where M is endowed with the

product symplectic structure) is also Poisson, with momentum mapping µ : M ’ g—

given by µ = µ1 —¦π1 + µ2 —¦π2 , where πi : M ’ Mi is the projection onto the i-th factor.

(ii) Show that if » : G—M ’ M is a Poisson action of a connected group G on a symplectic

manifold (M, „¦) with equivariant momentum mapping µ : M ’ g— and H ‚ G is

a (connected) Lie subgroup, then the restricted action of H on M is also Poisson

and the associated momentum mapping is the composition of µ with the natural

mapping g— ’ h— induced by the inclusion h ’ g.

E.7.2 126

(iii) Let (V, „¦) be a symplectic vector space and let G = Sp(V, „¦) ( Sp(n, R) where the

dimension of V is 2n). Show that the natural action of G on V is Poisson, with

momentum mapping µ(x) = ’ 1 x — (x „¦) . (Use the identi¬cation of g = sp(V, „¦)

2

—

with g de¬ned by the nondegenerate quadratic form a, b = tr(ab).) Show that the

mapping S 2 (V ) ’ sp(V, „¦) de¬ned on decomposables by

1

x—¦y ’ ’ x — (y „¦) + y — (x „¦)

2

is an isomorphism of Sp(V, „¦)-representations. Using this isomorphism, we can inter-

pret the momentum mapping as the quadratic mapping µ : V ’ S 2 (V ) de¬ned by

˜

the rule µ(x) = 1 x2 . What are the clean values of µ? (There are no regular values.)

˜ 2

Let Mk be the product of k copies of V and let G act ˜diagonally™ on Mk . Discuss the

clean values and regular values (if any) of µk : Mk ’ g— . What can you say about the

corresponding symplectic quotients? (It may help to note that the group O(k) acts

on Mk in such a way that it commutes with the diagonal action and the corresponding

momentum mapping.)

9. The Shifting Trick. It turns out that reduction at a general ξ ∈ g can be reduced

to reduction at 0 ∈ g. Here is how this can be done: Suppose that » : G — M ’ M is

a Poisson action on the symplectic manifold (M, „¦), that µ : M ’ g— is a corresponding

momentum mapping, and that ξ is an element of g— . Let M ξ , „¦ξ be the symplectic

product of (M, „¦) with (G·ξ, „¦ξ ) and let µξ : M ξ ’ g— be the corresponding combined

momentum mapping. (Thus, by the computation for coadjoint orbits done in the text,

µξ (m, a·ξ) = µ(m) ’ a·ξ.)

(i) Show that 0 ∈ g— is a clean value for µξ if and only if ξ is a clean value for µ.

Assume for the rest of the problem that ξ is a clean value of µ.

(ii) There is a natural identi¬cation of the G-orbits in (µξ )’1 (0) ‚ M ξ with the Gξ -

orbits in µ’1 (ξ) and that there is a smooth structure on G\(µξ )’1 (0) for which the

map π0 : (µξ )’1 (0) ’ G\(µξ )’1 (0) is a smooth submersion if and only if there is

ξ

a smooth structure on Gξ \µ’1 (ξ) for which the map πξ : µ’1 (ξ) ’ Gξ \µ’1 (ξ) is a

smooth submersion. In this case, the natural identi¬cation of the two quotient spaces

is a di¬eomorphism.

(iii) This natural identi¬cation is a symplectomorphism of (M ξ )0 , („¦ξ )0 with (Mξ , „¦ξ ).

This shifting trick will be useful when we discuss K¨hler reduction in the next Lecture.

a

10. Matrix Calculations. The purpose of this exercise is to let you get some practice

in a case where everything can be written out in coordinates.

Let G = GL(n, R) and let Q: gl(n, R) ’ R be a non-degenerate quadratic form.

Show that if we use the inclusion mapping x: GL(n, R) ’ Mn—n as a coordinate chart,

then, in the associated canonical coordinates (x, p), the Lagrangian L takes the form

L = 1 x’1 p, x’1 p Q . Show also that ωL = x’1 p, x’1 dx Q .

2

Now compute the expression for the momentum mapping µ and the Euler-Lagrange

equations for motion under the Lagrangian L. Show directly that µ is constant on the

solutions of the Euler-Lagrange equations.

E.7.3 127

Suppose that Q is Ad-invariant, i.e., Q Ad(g)(x) = Q(x) for all g ∈ G and x ∈ g.

Show that the constancy of µ is equivalent to the assertion that p x’1 is constant on the

solutions of the Euler-Lagrange equations. Show that, in this case, the L-critical curves in

G are just the curves γ(t) = γ0 etv where γ0 ∈ G and v ∈ g are arbitrary.

Finally, repeat all of these constructions for the general Lie group G, translating

everything into invariant notation (as opposed to matrix notation).

11. Euler™s Equation. Look back over the example given in the Lecture of left-invariant

metrics on Lie groups. Suppose that γ: R ’ G is an L-critical curve. De¬ne ξ(t) =

„Q ω(γ(t)) . Thus, ξ: R ’ g— . Show that the image of ξ lies on a single coadjoint orbit.

™

Moreover, show that ξ satis¬es Euler™s Equation:

ξ + ad— „Q (ξ) (ξ) = 0.

’1

™

The reason Euler™s Equation is so remarkable is that it only involves “half of the

variables” of the curve γ in T G.

™

Once a solution to Euler™s Equation is found, the equation for ¬nding the original

’1

curve γ is just γ = Lγ „Q (ξ) , which is a Lie equation for γ and hence is amenable to

™

Lie™s method of reduction.

Actually more is true. Show that, if we set ξ(0) = ξ0 , then the equation Ad— (γ)(ξ) = ξ0

determines the solution γ of the Lie equation with initial condition γ(0) = e up to right

multiplication by a curve in the stabilizer subgroup Gξ0 . Thus, we are reduced to solving

a Lie equation for a curve in Gξ0 . (It may be of some interest to recall that the stabilizer

of the generic element · ∈ g— is an abelian group. Of course, for such ·, the corresponding

Lie equation can be solved by quadratures.)

12. Project: Analysis of the Rigid Body in R3 . Go back to the example of the

motion of a rigid body in R3 presented in Lecture 4. Use the information provided in

the previous two Exercises to show that the equations of motion for a free rigid body

are integrable by quadratures. You will want to ¬rst compute the coadjoint action and

describe the coadjoint orbits and their stabilizers.

13. Verify that, under the hypotheses of Theorem 2, the dimension of the reduced space

Mξ is given by the formula

dim Mξ = dim M ’ dim G ’ dim Gξ + 2 dim Gm

where Gm is the stabilizer of any m ∈ µ’1 (ξ).

(Hint: Show that for any m ∈ µ’1 (ξ), we have

dim Tm µ’1 (ξ) + dim Tm G · m = dim M

and then do some arithmetic.)

E.7.4 128

14. In the reduction process, what is the relationship between Mξ and MAd— (g)(ξ) ?

15. Suppose that »: G — M ’ M is a Poisson action and that Y is a symplectic vector

¬eld on M that is G-invariant. Then according to Proposition 1, Y is tangent to each

of the submanifolds µ’1 (ξ) (when ξ is a clean value of µ). Show that, when the sym-

plectic quotient Mξ exists, then there exists a unique vector ¬eld Yξ on Mξ that satis¬es

Yξ πξ (m) = πξ Y (m) . Show also that Yξ is symplectic. Finally show that, given an

integral curve γ: R ’ Mξ of Yξ , then the problem of lifting this to an integral curve of Y

is reducible by “¬nite” operations to solving a Lie equation for Gξ .

This procedure is extremely helpful for two reasons: First, since Mξ is generally quite

a bit smaller than M, it should, in principle, be easier to ¬nd integral curves of Yξ than

integral curves of Y . For example, if Mξ is two dimensional, then Yξ can be integrated

by quadratures (Why?). Second, it very frequently happens that Gξ is a solvable group.

As we have already seen, when this happens the “lifting problem” can be integrated by (a

sequence of) quadratures.

E.7.5 129

Lecture 8:

Recent Applications of Reduction

In this Lecture, we will see some examples of symplectic reduction and its generaliza-

tions in somewhat non-classical settings.

In many cases, we will be concerned with extra structure on M that can be carried

along in the reduction process to produce extra structure on Mξ . Often this extra structure

takes the form of a Riemannian metric with special holonomy, so we begin with a short

review of this topic.

Riemannian Holonomy. Let M n be a connected and simply connected n-manifold,

and let g be a Riemannian metric on M. Associated to g is the notion of parallel transport

along curves. Thus, for each (piecewise C 1) curve γ: [0, 1] ’ M, there is associated a linear

mapping Pγ : Tγ(0) M ’ Tγ(1) M, called parallel transport along γ, which is an isometry of

’1

vector spaces and which satis¬es the conditions Pγ = Pγ and Pγ2 γ1 = Pγ2 —¦ Pγ1 where γ ¯

¯

is the path de¬ned by γ (t) = γ(1 ’ t) and γ2 γ1 is de¬ned only when γ1 (1) = γ2 (0) and, in

¯

this case, is given by the formula

for 0 ¤ t ¤ 1 ,

γ1 (2t) 2

γ2 γ1 (t) =

γ2 (2t ’ 1) for 2 ¤ t ¤ 1.

1

These properties imply that, for any x ∈ M, the set of linear transformations of the form Pγ

where γ(0) = γ(1) = x is a subgroup Hx ‚ O(Tx M) and that, for any other point y ∈ M,

we have Hy = Pγ Hx Pγ where γ: [0, 1] ’ M satis¬es γ(0) = x and γ(1) = y. Because we

¯

are assuming that M is simply connected, it is easy to show that Hx is actually connected

and hence is a subgroup of SO(Tx M).

´

Elie Cartan was the ¬rst to de¬ne and study Hx . He called it the holonomy of g at x.

He assumed that Hx was always a closed Lie subgroup of SO(Tx M), a result that was only

later proved by Borel and Lichnerowitz (see [KN]).

Georges de Rham, a student of Cartan, proved that, if there is a splitting Tx M =

V1 • V2 that remains invariant under all the action of Hx , then, in fact, the metric g is

locally a product metric in the following sense: The metric g can be written as a sum of the

form g = g1 + g2 in such a way that, for every point y ∈ M there exists a neighborhood U

of y, a coordinate chart (x1 , x2 ): U ’ Rd1 — Rd2 , and metrics gi on Rdi so that gi = x— (¯i ).

¯ ig

He also showed that in this reducible case the holonomy group Hx is a direct product

of the form Hx — Hx where Hx ‚ SO(Vi ). Moreover, it turns out (although this is not

1 2 i

obvious) that, for each of the factor groups Hx , there is a submanifold Mi ‚ M so that

i

i

Tx Mi = Vi and so that Hx is the holonomy of the Riemannian metric gi on Mi .

From this discussion it follows that, in order to know which subgroups of SO(n) can

occur as holonomy groups of simply connected Riemannian manifolds, it is enough to ¬nd

the ones that, in addition, act irreducibly on Rn . Using a great deal of machinery from the

theory of representations of Lie groups, M. Berger [Ber] determined a relatively short list

L.8.1 130

of possibilities for irreducible Riemannian holonomy groups. This list was slightly reduced

a few years later, independently by Alexseevski and by Brown and Gray. The result of

their work can be stated as follows:

Theorem 1: Suppose that g is a Riemannian metric on a connected and simply connected

n-manifold M and that the holonomy Hx acts irreducibly on Tx M for some (and hence

every) x ∈ M. Then either (M, g) is locally isometric to an irreducible Riemannian

symmetric space or else there is an isometry ι: Tx M ’ Rn so that H = ι Hx ι’1 is one of

the subgroups of SO(n) in the following table.

Irreducible Holonomies of Non-Symmetric Metrics

Subgroup Conditions Geometrical Type

SO(n) any n generic metric

U(m) n = 2m > 2 K¨hler

a

SU(m) n = 2m > 2 Ricci-¬‚at K¨hler

a

Sp(m)Sp(1) n = 4m > 4 Quaternionic K¨hler

a

Sp(m) n = 4m > 4 hyperK¨hler

a

G2 n=7 Associative

Spin(7) n=8 Cayley

A few words of explanation and comment about Theorem 1 are in order.

First, a Riemannian symmetric space is a Riemannian manifold di¬eomorphic to a

homogeneous space G/H where H ‚ G is essentially the ¬xed subgroup of an involutory

homomorphism σ: G ’ G that is endowed with a G-invariant metric g that is also invariant

under the involution ι: G/H ’ G/H de¬ned by ι(aH) = σ(a)H. The classi¬cation of the

Riemannian symmetric spaces reduces to a classi¬cation problem in the theory of Lie

algebras and was solved by Cartan. Thus, the Riemannian symmetric spaces may be

regarded as known.

Second, among the holonomies of non-symmetric metrics listed in the table, the ranges

for n have been restricted so as to avoid repetition or triviality. Thus, U(1) = SO(2) and

SU(1) = {e} while Sp(1) = SU(2), and Sp(1)Sp(1) = SO(4).

Third, according to S. T. Yau™s celebrated proof of the Calabi Conjecture, any compact

complex manifold for which the canonical bundle is trivial and that has a K¨hler metric

a

also has a Ricci-¬‚at K¨hler metric (see [Bes]). For this reason, metrics with holonomy

a

SU(m) are often referred to as Calabi-Yau metrics.

Finally, I will not attempt to discuss the proof of Theorem 1 in these notes. Even

with modern methods, the proof of this result is non-trivial and, in any case, would take

us far from our present interests. Instead, I will content myself with the remark that it

is now known that every one of these groups does, in fact, occur as the holonomy of a

Riemannian metric on a manifold of the appropriate dimension. I refer the reader to [Bes]

for a complete discussion.

L.8.2 131

We will be particularly interested in the K¨hler and hyperK¨hler cases since these

a a

cases can be characterized by the condition that the holonomy of g leaves invariant certain

closed non-degenerate 2-forms. Hence these cases represent symplectic manifolds with

“extra structure”, namely a compatible metric.

The basic result will be that, for a manifold M that carries one of these two structures,

there is a reduction process that can be applied to suitable group actions on M that preserve

the structure.

K¨hler Manifolds and Algebraic Geometry.

a

In this section, we give a very brief introduction to K¨hler manifolds. These are